Essential norm of the differential operator

TL;DR

This paper investigates the essential norm of the differential operator between generalized Fock spaces, extending previous spectral studies to growth type spaces and providing estimates for the essential norm in various cases.

Contribution

It extends prior spectral analysis of the differential operator to growth type Fock spaces and estimates its essential norm across all relevant exponent ranges.

Findings

Essential norm is comparable to the operator norm when the operator is not compact.

Derived explicit estimates for the essential norm of the differential operator.

Showed the essential norm is asymptotically equivalent to a specific function of parameters.

Abstract

This paper is a follow-up contribution to our work [10] where we studied some spectral properties of the differential operator $D$ acting between generalized Fock spaces $\mathcal{F}_{(m,p)}$ and $\mathcal{F}_{(m,q)}$ when both exponents $p$ and $q$ are finite. In this note we continue to study the properties for the case when at least one of the spaces is growth type. We also estimate the essential norm of $D: \mathcal{F}_{(m,p)}\to \mathcal{F}_{(m,q)}$ for all $1\leq p, q\leq \infty$, and showed that if the operator fails to be compact, then its essential norm is comparable to the operator norm and $\|D\|_e \simeq \big|m^{2+p}-m^{1+p}\big|^{\frac{1}{p}}\simeq \|D\|.$